Questions tagged [operator-theory]
Operator theory is the branch of functional analysis that focuses on bounded linear operators, but it includes closed operators and nonlinear operators. Operator theory is also concerned with the study of algebras of operators.
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Weighted shift operators with absolutely equal weights are unitarily equivalent
I have been stuck on the following exercise from Elementary Functional Analysis by Barbara MacCluer:
Exercise 2.10. Let $\{a_n\}_{n = 1}^\infty$ be a bounded sequence of complex numbers. Fix an ...
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Proof relies on the operator being a contraction...what is it in the statement of the lemma implies that the operator is a contraction?
I am trying to understand the following.
Lemma:
Let $H$ be a Hilbert space, let $\Phi = \{E_a\}$ be a linear map $B(H) \to B(H)$ where $\Phi(X) = \sum_a E_aXE_a^*$ for all $X \in B(H)$ that is ...
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Principal Symbol of the Fractional Laplacian on Manifolds
In the euclidean space $\mathbb{R}^n$, we can define the Fractional Laplacian as
$$(-\Delta)^s f := \int |\xi|^{2s}\hat{f}(\xi)e^{ix\cdot\xi}d\xi.$$
The principal symbol is clearly $p(x,\xi)=|\xi|^{2s}...
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Help understanding a proof about trace preserving and positive operators.
Edit: Going to try to make this better.
The implications (i) -> (ii) and (i) -> (iii) are clear to me. I'm not quite sure where the positive operators come from in (ii) -> (iii), but I had ...
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$T + \|T\| \cdot \bf 1$ is a positive operator?
I'm look at a proof where the author claims that we may assume that a self-adjoint operator $T$ is in fact positive by replacing $T$ with $T + \|T\| \cdot 1$...why can we do this?
Is $T + \|T\| \cdot \...
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Difficulty in understanding the proof of Theorem 2, section 40 of 'Introduction to Hilbert spaces and theory of spectral multiplicity' by Halmos
Suppose we are given a measurable space $(X, \Omega)$ where $\Omega$ is the $\sigma$-algebra of subsets of $X$ and a Hilbert space $H$. Let $E: \Omega \rightarrow B(H)$ be the spectral measure on $\...
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Reference request: uniformly continuous semigroups of nonlinear (Lipschitz) operators
Consider a Banach space $X$ with norm $\vert\cdot\vert$, and call an operator $A\colon X\to X$ Lipschitz whenever
$$\sup_{f\neq g} \frac{\vert Af-Ag\vert}{\vert f-g\vert}<+\infty;$$
the Lipschitz ...
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An invariant subspace for the adjoints of Kraus operators is a reducing subspace for the Kraus operators themselves.
This might be a long shot, but I am completely flummoxed.
I am trying to understand the proof of Theorem III.4. The theorem uses a lemma that states "an invariant subspace for the adjoints of ...
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Operator less than a projection with equal trace?
If $A$ is an operator on a Hilbert space $H$ and $P$ is a projection on $H$ with $0 \leq A \leq P$ and $\operatorname{Tr}(A) = \operatorname{Tr}(P)$, then $A = P$.
My proof:
Since $0 \leq A \leq P$, ...
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$T$, $S$ are closable implies that $\bar{T}\circ \bar{S}$ is closable
Let $T$ and $S$ be two unbounded closable densely defined operators on $H$. Assume additionally that $T\circ S$ is densely defined (and, if needed, closable and also that $\text{im}S \subset T$). Is ...
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Higson's homotopy invariance result
I am learning about operator algebras and $KK$-theory, a result I find very striking is the following :
Any split-exact $K$-stable functor $F : C^*\text{-alg} \to \text{Ab}$ is necessarily homotopy ...
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An interesting family of seminorms $\mathcal F$ and comparison between the topology generated by this seminorms and the Weak Operator Topology.
I am learning functional analysis and I am stuck with the following questions from Strong Operator Topology and Weak Operator topology on $\mathcal B(H)=\{T:H\to H:T$ is Op-Norm continuous,linear $\}$....
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Map defined by matrix units. [closed]
If $\{|\alpha_i\rangle\}_{i=1}^m$ and $\{| \beta_k \rangle\}_{k=1}^n$ are orthonormal bases of subsystems $A$ and $B$ respectively of a Hilbert space $H = A \otimes B$ and
$$P_{kl} = |\alpha_k\rangle\...
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If $A, B \in B(H)$ with $0 \leq A \leq PBP$ for $B \geq 0$, then $A = PAP$
Trying to prove this lemma:
Let $H$ be a Hilbert space and let $P \in B(H)$ be a projection.
If $A, B \in B(H)$ with $0 \leq A \leq PBP$ for $B \geq 0$, then $A = PAP$.
My first instinct is to look at
...
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Invertibility of tridiagonal operators
Are there any criteria or sufficient condition for the invertibility of tridiagonal operators in $l^2 (\mathbb{Z})$ ? This question arose from the following task — I needed to find the conditions for ...
